Calculus Examples

$(a (1 - \frac{w}{N}) - c r) w$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$\frac{d}{d w} [(a \cdot 1 + a (- \frac{w}{N}) - c r) w]$

Step 1.2

Multiply $a$ by $1$ .

$\frac{d}{d w} [(a + a (- \frac{w}{N}) - c r) w]$

Step 1.3

Rewrite using the commutative property of multiplication.

$\frac{d}{d w} [(a - a \frac{w}{N} - c r) w]$

Step 1.4

Combine $\frac{w}{N}$ and $a$ .

$\frac{d}{d w} [(a - \frac{w a}{N} - c r) w]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d w} [f (w) g (w)]$ is $f (w) \frac{d}{d w} [g (w)] + g (w) \frac{d}{d w} [f (w)]$ where $f (w) = a - \frac{w a}{N} - c r$ and $g (w) = w$ .

$(a - \frac{w a}{N} - c r) \frac{d}{d w} [w] + w \frac{d}{d w} [a - \frac{w a}{N} - c r]$

Step 3

Differentiate.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 1$ .

$(a - \frac{w a}{N} - c r) \cdot 1 + w \frac{d}{d w} [a - \frac{w a}{N} - c r]$

Step 3.2

Multiply $a - \frac{w a}{N} - c r$ by $1$ .

$a - \frac{w a}{N} - c r + w \frac{d}{d w} [a - \frac{w a}{N} - c r]$

Step 3.3

By the Sum Rule, the derivative of $a - \frac{w a}{N} - c r$ with respect to $w$ is $\frac{d}{d w} [a] + \frac{d}{d w} [- \frac{w a}{N}] + \frac{d}{d w} [- c r]$ .

$a - \frac{w a}{N} - c r + w (\frac{d}{d w} [a] + \frac{d}{d w} [- \frac{w a}{N}] + \frac{d}{d w} [- c r])$

Step 3.4

Since $a$ is constant with respect to $w$ , the derivative of $a$ with respect to $w$ is $0$ .

$a - \frac{w a}{N} - c r + w (0 + \frac{d}{d w} [- \frac{w a}{N}] + \frac{d}{d w} [- c r])$

Step 3.5

Add $0$ and $\frac{d}{d w} [- \frac{w a}{N}]$ .

$a - \frac{w a}{N} - c r + w (\frac{d}{d w} [- \frac{w a}{N}] + \frac{d}{d w} [- c r])$

Step 3.6

Since $- \frac{a}{N}$ is constant with respect to $w$ , the derivative of $- \frac{w a}{N}$ with respect to $w$ is $- \frac{a}{N} \frac{d}{d w} [w]$ .

$a - \frac{w a}{N} - c r + w (- \frac{a}{N} \frac{d}{d w} [w] + \frac{d}{d w} [- c r])$

Step 3.7

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 1$ .

$a - \frac{w a}{N} - c r + w (- \frac{a}{N} \cdot 1 + \frac{d}{d w} [- c r])$

Step 3.8

Multiply $- 1$ by $1$ .

$a - \frac{w a}{N} - c r + w (- \frac{a}{N} + \frac{d}{d w} [- c r])$

Step 3.9

Since $- c r$ is constant with respect to $w$ , the derivative of $- c r$ with respect to $w$ is $0$ .

$a - \frac{w a}{N} - c r + w (- \frac{a}{N} + 0)$

Step 3.10

Simplify terms.

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Step 3.10.1

Add $- \frac{a}{N}$ and $0$ .

$a - \frac{w a}{N} - c r + w (- \frac{a}{N})$

Step 3.10.2

Combine $w$ and $\frac{a}{N}$ .

$a - \frac{w a}{N} - c r - \frac{w a}{N}$

Step 3.10.3

Subtract $\frac{w a}{N}$ from $- \frac{w a}{N}$ .

$a - c r - 2 \frac{w a}{N}$

Step 3.10.4

Combine $- 2$ and $\frac{w a}{N}$ .

$a - c r + \frac{- 2 (w a)}{N}$

Step 3.10.5

Move the negative in front of the fraction.

$a - c r - \frac{2 w a}{N}$